Parameterization of invariant manifolds for periodic orbits (II): a-posteriori analysis and computer assisted error bounds

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1 Parameterization of invariant manifolds for periodic orbits II: a-posteriori analysis and computer assisted error bounds Roberto Castelli Jean-Philippe Lessard Jason D. Mireles James July 3, 216 Abstract In this paper we develop mathematically rigorous computer assisted techniques for studying high order Fourier-Taylor parameterizations of local stable/unstable manifolds for hyperbolic periodic orbits of analytic vector fields. We exploit the numerical methods developed in [1] in order to obtain a high order Fourier-Taylor series expansion of the parameterization. The main result of the present work is an a-posteriori Theorem which provides mathematically rigorous error bounds. The hypotheses of the theorem are checked with computer assistance. The argument relies on a sequence of preliminary computer assisted proofs where we validate the numerical approximation of the periodic orbit, its stable/unstable normal bundles, and the jets of the manifold to some desired order M. We illustrate our method by implementing validated computations for two dimensional manifolds in the Lorenz equations in R 3 and a three dimensional manifold of a suspension bridge equation in R 4. 1 Introduction The present work is the second paper in a series started in [1]. The purpose of this series is to study the partial differential equation 1 below. Our interest in Equation 1 is due to the fact that its solutions parametrize local stable/unstable manifolds associated with hyperbolic periodic orbits of ordinary differential equations. We focus on ordinary differential equations given by analytic vector fields, so the manifolds and hence the solutions of Equation 1 we consider are analytic. Paper I [1] is devoted to efficient numerical solution of the partial differential equation. Since solutions of Equation 1 parametrize embedded cylinders in phase space, it is natural to look for a formal solution expressed as a Fourier-Taylor series. The numerical scheme developed in Paper I exploits Floquet theory to find elementary recursion relations for the Fourier-Taylor coefficients of the formal series. These recursion relations can be rapidly solved to any desired order. VU University Amsterdam, Department of Mathematics, De Boelelaan 181, 181 HV Amsterdam, The Netherlands. r.castelli@vu.nl. Université Laval, Département de Mathématiques et de Statistique, Pavillon Alexandre-Vachon, 145 avenue de la Médecine, Québec, QC, G1V A6, CANADA. jean-philippe.lessard@mat.ulaval.ca. Florida Atlantic University, Department of Mathematical Sciences, Science Building, Room 234, 777 Glades Road, Boca Raton, Florida, 33431, USA. jmirelesjames@fau.edu 1

2 The present paper is concerned with convergence of the formal series just mentioned. Our main result is Theorem 2.4, an a-posteriori theorem which allows us to establish the existence of a true solution of Equation 1 supposing we have a good enough approximate solution. The theorem also provides mathematically rigorous C bounds between the approximate and true solution. The hypotheses of the theorem are checked via finite numerical computations. In order to illustrate the use and performance of our method we implement computer assisted convergence proofs for some example applications, namely we compute some two dimensional manifolds for the Lorenz system as well as some three dimensional manifolds for a suspension bridge equation. Both papers in this series are based on a functional analytic framework for studying invariant manifolds known as the Parameterization Method. The theoretical core of this method is developed in the work of [2, 3, 4, 5], and we refer the reader back to paper I for more complete discussion of the method and its literature. The interested reader may also want to consult the recent book of [6]. Presently we recall only the main philosophy of the Parameterization Method, which is that many of the smooth invariant manifolds of dynamical systems theory are characterized by an infinitesimal conjugacy, or invariance equation. The invariance equation, which often takes the form of a nonlinear operator on a Banach space, may be approximately solved on the digital computer using existing tools of numerical analysis. The operator equation may also be susceptible to a-posteriori analysis, that is once we obtain a good numerical approximation it may be possible to obtain mathematically rigorous error bounds via a computer assisted Newton-Kantorovich argument in the tradition of [7]. The reader interested in the Parameterization Method as a framework for computer assisted proof can consult the works of [8, 9, 1, 11, 12, 13] and the references discussed therein. We remark that the works just cited deal with stable/unstable manifolds of fixed points for maps/equilibria for differential equations, and also with invariant circles for area preserving maps and their bundles. The present work is a contribution in this vein, where we extend existing validated numerical methods based on the Parameterization Method to hyperbolic periodic orbits of differential equations. In Section 1.1 we refine the discussion above, and review as much of the Parameterization Method as is needed for the present work. In the section just mentioned we also outline the main steps of the validation argument developed in the remainder of the paper. Before moving on however, several remarks are in order. Remark 1.1 First order constraints. Solutions of Equation 1 below are dynamically meaningful only after the imposition of certain first order constraints. These constraints require that the periodic orbit, its stable or unstable Floquet exponents, and its stable or unstable normal bundles are known exactly. Here exactly is interpreted in the sense of validated numerics, that is we need numerical approximations of this data along with explicit mathematically rigorous error bounds. Since our goal is to validate parameterizations of local stable/unstable manifolds using Fourier spectral methods, we require the first order data is given as Fourier series. Indeed, the methods of the present work exploit complex analytic properties of the first order data and we actually require some knowledge about domains of analyticity of the periodic orbit and the stable/unstable bundles bounds on the size of a strip about the real axis in the complex plane into which the periodic functions can be extended analytically. In practice this information is not readily available and some preliminary work is required. Recent advances in computer assisted Fourier analysis of periodic solutions of analytic differential equations [14, 15] and computer assisted Floquet analysis [16, 17] allow us to obtain the desired representation of the first order data. The methods of the works just 2

3 cited also provide the lower bounds on the domain of analyticity mentioned in the previous paragraph. See Remark 2.9 for more detailed discussion of this point. Remark 1.2 Features of the Parameterization Method. Many functional analytic methods for proving stable manifold theorems are based on graph transform type arguments, Lyapunov-Perron operators, or sequence space arguments ala Irwin see a standard text on dynamical systems such as [18, 19, 2]. While it is possible to adapt such arguments for a- posteriori computer assisted analysis this approach has some draw backs, such as the need to compute the composition of the unknown parameterization with itself, and the requirement that the representation of the manifold is expressed as the graph of a function. The Parameterization Method on the other hand, requires only composition of the unknown function with the known vector field. Moreover there is no requirement that the parameterization be the graph of a function, hence it is possible to follow folds in the embedding. Another advantage is that the parameterization of the manifold satisfies a conjugacy relation which recovers the dynamics on the manifold in addition to the embedding. The price we pay is the appearance of some non-resonance conditions between the Floquet exponents of the periodic orbit. These non-resonance conditions have no analogue in the classical approaches mentioned in the first paragraph of this remark. However we must point out that a-the non-resonance conditions are satisfied generically that is for open sets in the parameter space of the vector field, and b-that it is possible to modify the underlying conjugacy relation in the Parameterization Method that is conjugate to a polynomial rather than a linear vector field in order to treat the resonant cases as well. We do not consider such degeneracies further in the present work. The interested reader can consult the works of [2, 3, 4] for more complete discussion of the resonant case, and can also see the work of [1] for computer aided proofs for resonant stable/unstable manifolds attached to equilibrium solutions of differential equations. Remark 1.3 Geometric methods: covering relations and cone conditions. Another approach to stable manifold theory is based on topological degree theory and cone conditions applied directly in the phase space. See for example the work of [21] on stable manifolds of fixed points of maps and the work of [22] on normally hyperbolic invariant manifolds. These geometric methods are well suited for adaptation to computer assisted proof, as is illustrated by the work of [23, 24, 25] see also the references discussed therein. Geometric methods have been used to give computer assisted proofs of a number of conjectures in celestial mechanics involving the existence and intersection of stable/unstable manifolds for periodic orbits [26, 27]. One of the advantage of the geometric methods is that they require only C 1 or C 2 assumptions on the vector field. Geometric methods also typically require only weak hyperbolicity assumptions, so for example there no nonresonance assumptions. The advantage of making weaker assumptions is seen in the work of [28], where the authors give computer assisted proofs of the existence and location of center manifolds, again in the context of a celestial mechanics problem. Of course the geometric methods result in only C 1 or sometimes only Lipschitz information about the manifold under consideration. Moreover these methods show the existence of the manifold often that the manifold is contained somewhere inside the union of a collection of polygons but do not explicitly recover the dynamics on the manifold in terms of a conjugacy. Quantitative properties of the embedding such as decay rates of jets or domain of analyticity seem to be unavailable using these methods. The geometric and analytic methods then complement one another, and the choice of method in a particular problem depends on the desired results. This situation not surprisingly mirrors the state of affairs in the classical qualitative pen and paper theory of 3

4 dynamical systems, where many important results in the theory have both geometric and functional analytic interpretations and proofs. Remark 1.4 Automatic differentiation for Fourier series. In the present work we implement our argument for vector fields with polynomial nonlinearities. This simplifies the technical details somewhat but is not a fundamental limitation. By exploiting ideas from automatic differentiation it is possible to apply the Fourier-Taylor methods of the present work to study analytic vector fields with nonlinearities given by elementary functions. For more complete discussion of automatic differentiation as a numerical tool in dynamical systems theory we refer to the works of [29, 3, 31, 32] and the references therein, though the list is by no means complete. The reader interested in automatic differentiation as a tool for computer assisted proof in Fourier analysis might consult the work of [15]. Remark 1.5 Computer assisted proof of connecting orbits. While the invariant manifolds studied in the present work are of interest in their own right, we also remark that intersections of stable/unstable manifolds play a central role in the global study of nonlinear systems. By studying the intersections of stable/unstable manifolds it is possible to learn about orbits which connect invariant sets to one another. In addition to being a critical component of Melnikov theory see for example the classical works of [33, 34, 35, 36] the intersections of stable/unstable manifolds explain global phenomena such as Arnold diffusion [37] and transport in celestial mechanics and fluid systems [38, 39, 4, 41, 42]. Of course this list of references does not even scratch the surface of the literature, and is only meant to point in the direction of further reading. Given the importance of connecting orbits in dynamical systems theory it is not surprising that many authors have developed numerical computational methods. We refer for example to the works of [43, 44, 45, 46, 47, 48] and the references discussed therein for much more complete discussion, though again this is by no means a complete list of references. Existence for connecting orbits occupies a central place in the computer assisted proof literature and we refer the interested reader to the works of [49, 5, 51, 52, 53, 54, 55, 9, 56, 8, 57] and the references discussed therein. Moreover we remark that the last two references just cited incorporate the Parameterization Method in a fundamental way, obtaining computer aided proofs of connecting orbits between equilibria of differential equations and fixed points of maps. A feature of the approach developed in these two papers is that the method of proof obtains the transversality of the connecting orbit for free that is if the method succeeds then the intersection of the stable/unstable manifolds are transverse. Combining the methods of the present work with a mathematically rigorous method for computer assisted analysis of boundary value problems as in [57] would be a natural extension, and will make the topic of a future study. The proposed method would give automatic transversality for connecting orbits between periodic orbits. 1.1 Review of the parameterization method for stable/unstable manifolds of periodic orbits In this section, and throughout the remainder of the paper, we assume the reader is familiar with classical stability analysis/floquet theory for periodic orbits of differential equations an excellent reference for this material is the book of [18]. We also remark that the material briefly reviewed here only provides some context for the work done below. The reader interested in more thorough discussion may also want to consult the works of [2, 3, 58, 59, 6] for more general coverage of the Parameterization Method, and [6, 61, 62, 4, 1] for more discussion of the Parameterization Method in the context of periodic orbits. 4

5 Now a little notation. With w C, let w = realw 2 + imagw 2 denote the complex absolute value. We endow C d with the max norm, so that if z = z 1,..., z d C d then Let z d = max 1 j d z j. def A r = {w C : imagw < r}, denote the complex strip of width r about the real axis. With T R, T > we say that the complex function γ : A r C d is T -periodic on A r if γw + T = γw, for all w A r. A function γ : A r C d is said to be analytic on A r if each component of γ is complex differentiable at each w A r. For k N, k 1 let D k ν def = { z = z 1,..., z k C k : z k < ν }, denote the poly-disk of radius ν about the origin in C k. Throughout the sequel we are interested in functions P : A r D k ν C d with d k + 1. We say that P is T -periodic in the first variable or simply that P is T -periodic if P w + T, z 1,..., z k = P w, z 1,..., z k, for all w, z 1,..., z k A r D k ν. We say that P is analytic on A r D k ν if each component of P is complex differentiable in each variable separately at each point w, z 1,..., z k A r D k ν. Let T >, f : C d C d be an analytic vector field, and fix the complex numbers λ 1,..., λ k C. We are interested in 2T -periodic, analytic functions P : A r D k ν C d solving the partial differential equation w P w, z 1,..., z k + k λ j z j P w, z 1,..., z k = f[p w, z 1,..., z k ]. 1 z j j=1 When properly constrained, Equation 1 has special dynamical significance for the vector field f. In order to make this precise we state the following assumptions. Assumption 1. Suppose that γ : A r C d is an analytic, T -periodic solution for the vector field f, that is d γw = f[γw], dw and γw + T = γw for all w A r. Assumption 2. Suppose that λ 1,..., λ k are the stable or respectively unstable Floquet exponents for γ. Suppose in addition that these are all of the stable respectively unstable exponents, so that any remaining exponents are unstable or neutral respectively stable or neutral. Suppose now that ξ j : A r C d are analytic, 2T -periodic functions parameterizing the stable respectively unstable normal bundle of γ. More precisely suppose that realλ j <, for 1 j k respectively that realλ j >, for 1 j k and that ξ j w + 2T = ξ j w for all w A r and solve the eigenvalue problem d dw ξ jw + λ j ξ j w Df[γw]ξ j w =, for 1 j k. 5

6 Remark 1.6 Orientation of the stable/unstable normal bundles. The functions ξ j w are 2T -periodic as the stable/unstable bundles need not be orientable. Moreover, we can find a T -periodic basis function ξ j w if and only if the associated bundle is orientable. We now recall several properties enjoyed by solutions of Equation 1. P x x P W s loc W s loc w, z L Figure 1: Cartoon of the flow conjugacy Equation 4 satisfied by solutions of Equation 1. Claim 1. Flow conjugacy Let φ: R C d C d denote the flow generated by f. In fact we only need that φ is defined in a neighborhood of γ. Suppose that P : A r D k ν C d is an analytic 2T -periodic solution of Equation 1 which satisfies the first order constraints and P w,,..., = γw, 2 z j P w,,..., = ξ j w, for 1 j k. 3 Then the image of P is a local stable respectively unstable manifold for γ. In fact, P satisfies the flow conjugacy relation φ [P w, z 1,..., z k, t] = P w + t, e λ1t z 1,..., e λ kt z k, 4 for all t respectively t, that is the dynamics on the local manifolds parameterized by P are conjugate to the linear flow L: A r D k ν A r D k ν given by Lw, z 1,..., z n, t def = w + t, e λ1t z 1,..., e λkt z k. 6

7 The geometric meaning of this conjugacy is illustrated in Figure 1. For the elementary proof see Theorem 2.6 in [1]. Remark 1.7 Real vector fields, orbits, bundles, and parameterizations. The case of a real analytic vector field is of special interest, that is when x R d C d implies that fx R d. In this case we are especially interested in real analytic periodic orbits, that is γ : A r C d having that γt R d when t R. If γ is real analytic and λ j R for 1 j k then the basis functions ξ j can be chosen real analytic. In this case the solution P of Equation 1 can be taken real analytic. Another case of interest is that λ j and λ j+1 are a complex conjugate pair. In this case the associated basis functions ξ j and ξ j+1 can be taken as complex conjugates and arrange that P maps associated complex conjugate variables into R d. In other words, P is no longer real analytic, but there is a canonical method for obtaining the real image of P and hence the real stable unstable manifold associated with γ R d. See [1] for more complete discussion. Claim 2. Non-uniqueness Solutions of Equation 1 are not unique. Indeed suppose that P is a solution of Equation 1 constrained by Equations 2 and 3, and consider any collection Γ = {τ 1,..., τ k } of non-zero positive real scalars. Define the disk D k Γ,ν def = and the function Q: A r D k Γ,ν Cd by {z 1,..., z k C k : z j ν τ j and 1 j k }, Qw, z 1,..., z s def = P w, τ 1 z 1,..., τ k z k. By differentiating and evaluating at zero we see that Qw,,..., = P w,,..., = γw, and that Moreover Qw,,..., = τ j P w,,..., = τ j ξ j w. 5 z j z j f[qw, z 1,..., z k ] = f[p w, τ 1 z 1,..., τ k z k ] = w P w, τ 1z 1,..., τ k z k + = w Qw, z 1,..., z k + k λ j τ j z j P w, τ 1 z 1,..., τ k z k z j j=1 k λ j z j Qw, z 1,..., z k, z j as P is a solution of Equation 1. Then Q is an 2T periodic solution of Equation 1 on A r D k Γ,ν, satisfying constraint Equations 2 and 3 with a rescaled choice of basis functions for the normal bundle. Since P is analytic so is Q. If τ j < 1 for 1 j k then Q is also a solution of Equation 1 on A r D k ν, hence the non-uniqueness. If the τ j > 1 then the question of whether or not Q is a solution on A r D k ν is a question we return to momentarily, indeed it is one of the main concerns of the present work. Remark 1.8 Rescaling the basis of the stable/unstable normal bundle. Equation 5 shows that rescaling the domain by Γ leads to a corresponding rescaling of the basis functions for the stable respectively unstable normal bundle of the periodic orbits γ. In j=1 7

8 fact this is the only source of non-uniqueness in the problem, that is once the scalings of the ξ j are fixed then the solution of Equation 1, if it exists, is unique. See Claim 3 below and also the discussion in Section 5 of [4]. We remark that the freedom in the choice of the scaling of the basis functions can be exploited in numerical computations. Numerical implications of non-uniqueness were discussed in [1], and will play a role in the sequel. Suppose now that we look for a solution P of Equation 1 as a Taylor series P w, z = a wz. = Here =,..., k N k is the k-dimensional multi-index, = k, z = z 1,..., z k D k ν, and z = z z k k, and for each Nk the functions a : A r C d are analytic, 2T -periodic functions. Imposing the first order constraints 2 and 3 leads to a w = γw and a ej w = ξ j w, where =,..., N k and e j =,..., 1,..., is the j th vector of the canonical basis of R k. Plugging the Taylor expansion into Equation 1 and matching like powers of z shows that the periodic functions a : A r C d must solve the homological equations d dw a w + 1 λ k λ k a w Dfγwa w = R w, 6 for 2. Here R is a function only of the a β with β <, that is the homological equations are inhomogeneous linear differential equations which can be solved recursively to any order. Computation of the a is illustrated for a number of specific example problems in [1], and one sees that for a given vector field f the functions R w can be worked out explicitly. Definition 1.9. We say that the Floquet exponents λ 1,..., λ k are non-resonant if 1 λ k λ k λ j, 7 for each 2 and 1 j k. Note that since the real part of the λ j are all negative respectively positive this reduces to only a finite number of conditions, that is for large enough the non-resonance condition is automatically met. Claim 3. If λ 1,..., λ k are non-resonant then a exists and is unique for all 2. This is due to the fact that the homological equations given by 6 are linear with periodic coefficients. Hence the classical Floquet theorem gives that these equations have unique analytic 2T periodic solution assuming that Equation 7 holds. See also Section 2.3 of [1]. Only the convergence of the formal solution is in question. For fixed choice of ξ j w, 1 j k assume that the Floquet exponents λ 1,..., λ k are non-resonant and let P w, z = a wz, = be the associated formal solution of Equation 1. By Claim 1 we obtain another formal 8

9 solution by Qw, z def = P w, τ 1 z 1,..., τ k z k = a wτ 1 z 1,..., τ k z k 1,..., k = = = a wτ τ k k z. Moreover, as seen in Claim 1, this rescaling of the domain is equivalent to rescaling the basis of the stable respectively unstable normal bundle. Since by Claim 3 the coefficients with 2 are uniquely determined we see that: given the Taylor coefficients {a w} = of a particular formal solution of Equation 1 all other formal solutions are of the form for some choice of the scalars τ 1,..., τ k. q w = τ τ k k a w, 8 Claim 4. A-priori existence for small scalings Fix first order data γ, ξ j : A r C d for 1 j k, and let τ j = sup w A r ξ j w d and s = max 1 j k {τ j}. Suppose we fix also a ν >. The results of [2, 4] give that: if the Floquet multipliers are non-resonant then there exists an ɛ > so that for all s ɛ the formal solution associated with this choice of first order constraints converges, that is for small enough s the solution P : A r D k ν C d of Equation 1 subject to these constraints exists. The solution is unique, again up to the choice of the scalings ξ 1 w,..., ξ k w. Then for different choices of τ j ɛ we parametrize larger or smaller portions of the local stable respectively unstable manifold of the periodic orbits. The dependence of the Taylor coefficients on the scalings illustrated in Equation 8 make it clear that by choosing larger scalings for the basis functions ξ j w we obtain slower convergence of the series P. Hence for a fixed domain disk D k ν larger scalings correspond to a larger image of the parameterization in phase space. On the other hand, smaller choice of scalings make it more likely that the formal solution converges to a true solution. This is the fundamental balancing act inherent in the Parameterization Method: namely we want the image of P as large as possible in phase space so that the series still converges. Assumption 3. Suppose that the stable respectively unstable Floquet exponents λ 1,..., λ k are non-resonant. Choose τ 1,..., τ k > and suppose that the basis functions ξ 1 w,..., ξ k w are scaled so that τ j = sup w A r ξ j w d. Fix N N with N 2. For 2 N assume that a : A r C d are analytic, 2T -periodic solutions of the Homological Equation 6. Define the approximate solution P N : A r C k C d of Equation 1 by P N w, z = N = a wz. 9 9

10 Main question Existence and approximation: With the scalings τ 1,..., τ k, N 2, and P N w, z as in Assumption 3, let Hw, z = =N+1 a wz, be the tail function defined by the formal series solutions of the homological equations. For a particular choice of ν >, does H converge on A r D k ν? If so, can we bound H on A r D k ν? The remainder of the paper is organized as follows. In Section 2 we develop the main result of the present work. This is Theorem 2.4 which, when its a-posteriori hypotheses are satisfied, answers the existence and approximation questions at the same time. Section 3 deals with computer assisted validation of the solutions a w of the homological equations for 2 N, that is with obtaining the data postulated in Assumption 3. Sections 4 and 5 are devoted to applications: validation of manifolds with one and two stable/unstable Floquet exponents respectively. Appendix A contains some technical details associated with the application problem of Section 5. 2 A-posteriori analysis of Equation 1 We begin somewhat informally in order to introduce the main idea of the argument. Suppose that f : C d C d is an analytic vector field and that T, γw, λ 1,..., λ k, ξ 1 w,..., ξ k w, and P N w, z are as in Assumptions 1, 2 and 3 of Section 1.1. We seek a 2T -periodic function H : A r D k ν C d so that is an exact solution of Equation 1 for all w, z A r D k ν. Plugging P from Equation 1 into Equation 1 gives where f[p N w, z + Hw, z] = P w, z = P N w, z + Hw, z, 1 w [P N w, z + Hw, z] + D z [P N w, z + Hw, z]λz, 11 Λ def = λ λ k is the k k diagonal matrix with the Floquet exponents on the diagonal and zeros elsewhere. We expand the vector field to second order about the image of the approximate solution P N so that the left hand side of 11 becomes f[p N w, z + Hw, z] = f[p N w, z] + Df[P N w, z]hw, z + R[P N w, z, Hw, z]. 12 Here R is the second order Taylor remainder associated with the vector field f at the point P N w, z. Define the a-posteriori error function E N : A r D k ν C d associated with P N to be the function given by, E N w, z def = f[p N w, z] w P Nw, z D z P N w, zλz. 13 1

11 Plugging 12 and 13 into Equation 11 and rearranging gives w Hw, z + D zhw, zλz Df[P N w, z]hw, z = E N w, z + R[P N w, z, Hw, z]. We introduce the linear operator and rewrite 14 as 14 L[H]w, z def = w Hw, z + D zhw, zλz Df[P N w, z]hw, z, 15 Assuming that L is invertible gives L[H]w, z = E N w, z + R[P N w, z, Hw, z]. Hw, z = L 1 E N w, z + R[P N w, z, Hw, z], and we introduce the nonlinear operator Φ[H]w, z def = L 1 [E N w, z + R[P N w, z, Hw, z]. 16 We see that H is the truncation error associated with P N if and only if H is a fixed point of Equation 16. The remainder of the section is devoted to the study of Φ. We want to show that Φ has a unique fixed point in a small neighborhood of P N, and we solve the problem in three steps. Step 1: Show that the linear operator L is invertible. We will see that invertibility of L follows if N is large enough. Step 2: Establish some quadratic estimates for the nonlinear function R[P N w, z, Hw, z] given by Equation 12. Step 3: Obtain that the operator Φ is a contraction mapping in a certain neighborhood U δ. In order to formalize these steps we need to define appropriate Banach space norms. 2.1 Background: analytic functions and 2T -periodic families of analytic N-tails Let Mat m n C denote the collection of all m n matrices with complex entries. A Mat m n C we employ the norm A M = max 1 i m j=1 n a ij, where a ij is the usual complex absolute value. Let U C be a simply connected open domain in the complex plane, and f : U C an analytic function. We say that f is bounded on U if sup fw <. w U The set of all bounded analytic functions on U is a Banach space under the norm For f U def = sup fw. w U 11

12 The set of all functions f = f 1,..., f d such that each f j : A r C, 1 j d is analytic, bounded, and T -periodic is a Banach space under the product space maximum norm f r def = max f j A 1 j d r. Similarly, we say that P : A r D k ν C d is bounded if max P j A <. 1 j d r D k ν For P : A r D k ν C d given by the Taylor series P w, z 1,..., z k = = a wz z k k, with a : A r C d bounded analytic functions, we define the two norms P r,ν P r,ν def = a r ν, = def = max P j A. 1 j d r D k ν Note that it is always the case that P r,ν P r,ν even when the latter quantity is infinite. Now for 1 i, j N let a ij : A r C be bounded and 2T -periodic analytic functions, and consider the matrix Define the norm Aw = A r a 11 w... a 1N w..... a N1 w... a NN w def = max 1 i N. N a ij A r. 17 Then if g : A r C d is a bounded, analytic, T -periodic function then we obtain a new function Ag : A r C d by the matrix multiplication Awgw and have that j=1 Ag r A r g r. We say that an analytic function h: D k ν C is an analytic N-tail if h = z h =, for all Nk, N. Note that an analytic N-tail has Taylor series hz = =N+1 h z, converging absolutely and uniformly for z < ν. Analytic N-tails enjoy the following estimate. 12

13 Lemma 2.1. Suppose that h: D k ν C is an analytic N-tail with h D k ν = sup hz <. z D k ν Fix λ 1,..., λ k C and suppose that λ j µ < 1 for 1 j k for some µ >. Then sup hλ 1 z 1,..., λ k z k µ N+1 sup hz. 18 z D k ν z D k ν An elementary proof of 18 is given in [9]. The estimate above extends trivially to functions h: D k ν C d whose component functions are analytic N-tails. In the sequel we are interested in analytic N-tails with coefficients which are 2T -periodic on A r and analytic on A r D k ν. Such an H : A r D k ν C d is given by Hw, z = =N+1 h wz, where the sum converges absolutely for imagw < r, z < ν. Then for each fixed w A r the analytic function Hw, z is an analytic N-tail in z. We call such a function H a 2T - periodic family of analytic N-tails. The space of 2T -periodic families of analytic N-tails is a Banach space under both the r,ν and r,ν norms. Suppose that λ 1,..., λ k satisfy the hypothesis of Lemma 2.1 and let Λ be the diagonal matrix with λ 1,..., λ k as diagonal entries and zeros elsewhere. Then H r,ν < implies Hw, Λz r,ν µ N+1 H r,ν, 19 as Lemma 2.1 applies uniformly for each fixed w A r. For bounded analytic functions f : D k ν C d we have the following bounds on derivatives. The result is standard and we refer to [8] for the proof. Lemma 2.2 Cauchy Bounds. Suppose that f : D k ν C k C d is bounded and analytic. Then for any < σ 1 we have that i f νe σ 2π νσ f ν. 2.2 Validation values and the main theorem Take f : C d C d, T, N, γ : A r C d, λ 1,..., λ k C, ξ j : A r C d for 1 j k, a : A r C d for 2 N, and P N : A r C k C d as in Assumptions 1, 2, 3 of Section 1.1. We define the tube of radius ρ about γ to be U ρ γ = w A r D d ρ[γw]. Suppose that f is bounded and complex analytic on the tube U ρ γ C n. Consider where PN 1 def w, z = N =1 a wz. Let Df[P N w, z] = Df[γw + P 1 Nw, z], Ãw, z def = Df[P N w, z] Df[γw]. 13

14 For the applications considered in the present work f is always an M-th order polynomial with M = 2, 3. Then, since P N is an N-th order polynomial in z we have that Ãw, z = M 1N =1 A wz 2 is an M 1N-th order polynomial with 2T -periodic matrix coefficients A w. The explicit form of the A w is a problem dependent computation illustrated in examples. Now fix ν >. The following definition tabulates the constants which need to be computed in order to obtain validated bounds on the tail of P N. Definition 2.3 Validation values for an approximate solution of Equation 1. The positive constants κ, C, µ, ρ, ρ, M 1, M 2, and ɛ are called validation values for the approximate parameterization P N if i Dfγ r κ, 21 ii exp M 1N =1 A r 1 λ k λ k ν C, 22 where the A = A w are the coefficients of Ãw, z as given in 2, iii iv v µ min 1 i k realλ 1,..., realλ k, 23 max # { k, l 1 k, l n such that k l f j } M 1, 24 1 j n sup max max z U ργ 1 i n β =2 2 z β f iz M 2, 25 vi < ρ < ρ and N a r ν ρ, 26 =1 insuring that imagep N U ρ γ. vii Finally assume that where E N w, z is as defined in 13. E N r,ν ɛ, 27 Theorem 2.4 A-Posteriori Validation of a High Order Approximation of the Stable Manifold of a Periodic Orbit. Suppose that κ, C, µ, ρ, M 1, M 2, and ɛ are validation values for P N. Assume that N N and δ > have that N + 1 > κ µ, 28 14

15 { } N + δ < e 1 1µ κ min, ρ ρ 2nπM 1 M 2 C, 29 2 C ɛ < δ. 3 N + 1µ κ Then there is a unique periodic family of analytic N-tails H : A r D k ν C n with having that H r,ν δ, P w, z = P N w, z + Hw, z, is the exact solution of Equation 1 on A r D k ν. 2.3 Spaces and Lemmas First, we will exploit the following bound. Lemma 2.5. Suppose that κ, C are real positive constants with Dfγ r κ and M 1N exp A r 1 λ k λ k ν C. 31 =1 Then for any w A r, z D k ν, and t we have the bound t exp Df P N w + s, e Λs z ds Ce κt. Proof. Letting Ñ = M 1N note that t t Ãw + s, e Λs z ds Ñ A w + s e Λs z ds = = t =1 =1 Ñ A w + s e 1λ kλ k s z ds =1 Ñ t A r Ñ A r =1 Ñ =1 e 1µ kµ k s ds ν 1 e 1µ kµ k t 1 ν 1 µ k µ k A r 1 µ k µ k ν. 15

16 Then t exp Df P N w + s, e Λs z t t ds = exp Dfγw + s ds + Ã w + s, e Λs z ds t t exp Dfγw + s ds exp Ãw + s, e Λs z ds Ñ e κt exp A r 1 µ k µ k ν Ce κt. =1 Remark 2.6 General vector fields. If f is not a polynomial then of course the bound for C will be more complicated. We can always obtain an expression similar to Equation 31, plus a small remainder, by considering a Taylor expansion for f about any convenient point. On the other hand, if the nonlinearity of f is given by elementary functions, then it will be possible to bound C using techniques of automatic differentiation. We now define our function space. Let X def = { H : A r D k ν C d : H is a 2T -periodic family of analytic N-tails and H ν,r < }, and recall that X is a Banach Space under the r,ν norm. We denote the closed delta neighborhood of the origin in X by U δ = { H X : H r,ν δ }. Lemma 2.7. Let L as defined by Equation 15. If then L is boundedly invertible on X and L 1 X N + 1 > κ µ, 32 C N + 1µ κ. 33 Proof. Let S X. We will show the existence of H X such that L[H] = S. Assuming it exists it does and it is given by Equation 37 below, consider the equation L[H]w, z = w Hw, z + D zhw, zλz Df[P N w, z]hw, z = Sw, z. 34 For t R + we fix w A r, z D k ν and define the curve Γ : R + C k+1 given by w + t Γt = e Λt. z We now make the change of variables Ht def = H Γt = H w + t, e Λt z, Ψt def = Df[P N Γ]t = Df[P N w + t, e Λt z], St def = S Γt = S w + t, e Λt z, 16

17 and observe that d dt Ht = DHΓt Γ t = [ w HΓt D z HΓt] 1 e Λt Λz = w H w + t, e Λt z + D z H w + t, e Λt z e Λt Λz. Then we consider the ordinary differential equation d dt Ht Ψt Ht = St, and note that H is the solution of Equation 34. We introduce the integrating factor and note that Ω = I and Then for any t, a R we have that Ωt def = e t Ψτ dτ, Ω 1 t = e t Ψτ dτ. Ωt Ht = Ωa Ha + t a Ωβ Sβ dβ. 35 The next step is to evaluate the limit as a goes to infinity. We assume that H is bounded an assumption which will be justified momentarily and see recalling the definition of µ in 23 that Ωa Ha = e a Ψτ dτ Hw + a, e Λa z a e a Ψτ dτ e µ a N+1 Hw + a, z b Ce κa e N+1µ a H r,ν = C H r,νe [ N+1µ +κ ]a, 36 where a follows from Lemma 2.1 and b follows from Lemma 2.5. Now the hypothesis of Equation 32 implies that N + 1µ > κ and we have lim a Ωa Ha = C H r,ν lim a e[ N+1µ +κ ]a =. Then at least formally we have that Equation 35 becomes Taking t = gives Hw, z = H = Ht = Ω 1 t t Ωβ Sβ dβ. [e β Df[P Nw+τ,e Λτ z ] dτ ] S w + β, e Λβ z dβ. 37 Now we take Equation 37 as the definition of H, and by running the argument backwards we see that H solves Equation 34, assuming that we can show that H is bounded. 17

18 It is also clear upon inspection that H so defined is T -periodic on A r in the variable w, owing to the fact that P N and S are. What remains is to show that H is bounded and analytic in both w and z and that H is a T -periodic family of analytic N-tails. By arguing as in Equation 36 we have Ωβ Sβ Ce [ N+1µ +κ]β S r,ν, 38 with S bounded and N + 1µ + κ < by hypothesis. It follows that Hw, z = H = Ωβ Sβ dβ C S r,ν e [ N+1µ +κ]β dβ C S r,ν N + 1µ κ, 39 and we see that H is indeed bounded. To see that H is analytic in any of w, z 1,..., z k we remark that Equation 38 shows that the integrand is bounded and we know the integrand is analytic in each variable. Morera s Theorem [63] then gives the analyticity of H. To see that H is an N-tail we simply use that S is an N-tail and apply the Leibnitz rule on Equation 37 exploiting the boundedness of the integrand in order to pass the derivative under the integral. Since S was arbitrary we see that the desired inverse operator is defined explicitly by L 1 [S] = H. Taking the supremum in Equation 39 over all S with norm one gives the bound claimed in the Equation 33. Lemma 2.8. Assume that the hypotheses of Theorem 2.4 are satisfied. Given H X, let R: A r D k ν C d be the function defined by f[p N w, z + Hw, z] = f[p N w, z] + Df[P N w, z]hw, z + R[P N w, z, Hw, z]. 4 Then R is a T -periodic family of analytic N-tails satisfying the following bounds: For any δ ρ ρ and H U δ, For any δ ρ ρ e 1 and H 1, H 2 U δ we have RP N, H r,ν M 1 M 2 δ R[P N, H 1 ] R[P N, H 2 ] r,ν 2πedM 1M 2 δ H 1 H 2 r,ν. 42 Proof. Let s def = ρ ρ. For any fixed w, z A r D k ν and η D d s we have that f[p N w, z + η] = f[p N w, z ] + Df[P N w, z ]η + R[P N w, z, η], where R is given by the Lagrange form of the Taylor Remainder R j [P N w, z, η] = β =2 2 β! ηβ 1 1 t 2 η β f jp N w, z + tη dt

19 Applying Morera s Theorem e.g. see [63] to Equation 43 shows that R is analytic as w, z A r D k ν as well as η D d s vary. It is also clear that R j is T -periodic in w. Moreover R j [P N w, z, η] β =2 2 β! sup z U ργ 2 z β f jz η 2 d M 1 M 2 η 2 d. Fixing H U δ with δ < s we define R: A r D k ν C d by R[P N w, z, Hw, z] and have R[P N, H] r,ν M 1 M 2 δ 2, as desired. We now introduce a loss of domain parameter σ, 1] which is used to leverage the supremum bound on R into a Lipschitz bound. So, fix w, z A r D k ν and η 1, η 2 D d e σ s. Then R[P N w, z, η 1 ] R[P N w, z, η 2 ] = D η R[P N w, z, η]η 1 η 2, for some η D d e σ s. The fact that R is analytic and zero to second order in η implies that / η j R i is analytic and zero to first order in η for each 1 i, j d. Then for any < t < 1 we have D η R[P N w, z, tη] M t D η R[P N w, z, η] M, Let t = δ/se σ with < σ 1 and note that < t < 1 by the hypothesis given by Equation 29 of Theorem 2.4. It follows that D η [P N w, z, tη] M δ se σ D ηr[p N w, z, η] M δ sup DR[P N w, z, η] M se σ η =e σ s δeσ s 2πd sσ sup R[P N w, z, η] η =s δ 2πdeσ σs 2 M 1M 2 s 2 2πedM 1 M 2 δ, 44 as e σ /σ is minimized when σ = 1. Note that we have used the Cauchy Bounds of Lemma 2.2 in order to pass from the second to the third line of the estimate. The use of the Cauchy Bounds explains the loss of domain and the parameter σ appearing in the argument. Fixing H 1, H 2 U δ we now have R[P N, H 1 ] R[P N, H 2 ] r,ν 2πedM 1 M 2 δ H 1 H 2 r,ν. 2.4 Proof of Theorem 2.4 We will show that the operator Φ: X X defined in 16 has a unique fixed point in U δ X. First note that Φ is well defined as the hypothesis of Equation 28 allow us to apply Lemma 2.7 and obtain that L is boundedly invertible on X. 19

20 Now for any H U δ and consider Φ[H] r,ν = L 1 [E N + R[P N, H]] r,ν L 1 r,ν E N r,ν + L 1 r,ν R[P N, H] r,ν C ɛ + M1 M 2 δ 2 N + 1µ κ δ. 45 Here we have applied the bounds of Lemma 2.7, as well as the bound given by Equation 41, the the definition of ɛ from condition vii if the definition of validation values, as well as the hypothesis given by Equations 29 and 3 of the present theorem. This shows that in fact Φ : U δ U δ. What remains is to show that Φ is a contraction on U δ. To see this let H 1, H 2 U δ. The inequality hypothesized by Equation 29 of the present theorem gives that δ ρ ρ /e. It follows that we can apply the bound given by Equation 42 to the expression Φ[H 1 ] Φ[H 2 ] r,ν = L 1 R[P N w, z, H 1 w, z] R[P N w, z, H 2 w, z] r,ν C N + 1µ κ 2πedM 1M 2 δ H 1 H 2 r,ν. 46 Here we have again used the bounds of Lemma 2.7. Moreover, C N + 1µ κ 2πedM 1M 2 δ < 1 by the inequality hypothesised in Equation 29 of the present theorem. Then Φ: U δ U δ is a contraction mapping and hence has unique fixed point H U δ. Since H is the truncation error we have P P N r,ν = H r,ν δ, as desired. 2.5 Validation Algorithm The parameters ν and the scalings of the basis functions ξ j, 1 j k are free in the discussion above. In theory there is no difference between fixing the norms of the basis vectors and then choosing the size ν > of the validation domain, or vice versa. In practice however it is preferable to fix ν = 1 and choose numerically convenient scalings for the basis functions. This is simply due to the fact that ν = 1 is the most stable choice in the ν terms appearing in the Taylor-norms. Algorithms for optimizing the choice of the scalings are discussed in detail in [64], however we make a few heuristic comments. In many applications it is sufficient to simply compute once the parameterization P N with any convenient choice of scalings, and then examine the growth of the resulting p r. If these coefficients grow either too fast or too slow then simply rescale so that the desired decay rate is achieved. The dependence of the new decay on the choice of rescaling is given explicitly in Equation 8. A good heuristic is to choose that the highest order coefficients have magnitude close to machine epsilon. Once the scalings and ν are picked then we try the following algorithm. If the algorithm fails then we modify the scalings and try again. Recipe: Input choice of ν from above. 2

21 1. Choose: any positive constant κ satisfying 21. In practice κ will be an interval arithmetic bound on Dfγ r as this quantity depends only in the known Fourier coefficients of γ as well as the known decay rate bounds. 2. Choose: a positive constant C satisfying 22. In practice C is any bound on the sum obtained using interval arithmetic. 3. Choose: a constant ρ satisfying 26. In practice the terms a r are bound using the a r norms. 4. Choose: s >. For polynomial vector fields the choice of s is more or less arbitrary and we often obtain good results with s =.1ρ. Then define ρ to be any number with ρ + s ρ. If f is not polynomial then we have to pick a ρ > and estimate the second derivative of f in a ρ neighborhood of γ. This could be done either by hand, or more likely with some preliminary interval arithmetic. Note that if f has poles then this imposes a theoretical limit on the size of ρ. Next we make sure that ρ defined above is smaller than ρ. s is then the difference. 5. Compute: M 1 satisfying 24 and M 2 satisfying 25 on the tube U ρ γ. For quadratic vector fields the second partial derivatives are of course constant. 6. Choose: ɛ to be any positive number satisfying Choose: µ satisfying 23, that is any lower bound on the absolute values of the real parts of the stable eigenvalues. 8. Check: that condition 28 is satisfied. If not then the proof fails. 9. Choose: δ > so that 3 is satisfied. 1. Check: that 29 is satisfied. If not then the proof fails. 11. Return: δ. If the proof does not fail before Step 11, then it succeeds and we have the validated bound H r,ν H r,ν δ, for the truncation error associated with the approximation P N on the domain A r D k ν. Remark 2.9 Implementation details. In practice we must obtain the data hypothesized in Assumptions 1, 2 and 3 of Section 1.1 by separate computer assisted arguments. Since the functions γw, ξ 1 w,..., ξ k w, and a w for 2 N are analytic and periodic, it is reasonable to represent these functions as Fourier series. Then the first step is to compute numerical approximations γ M w = ξ M j w = a M w = M m= M M m= M M m= M γ m e 2πim T w, ξ j m e 2πim 2T w, 1 j k, a,m e 2πim 2T w, 2 N. 21

22 In addition to these numerical approximations we need positive constants r, ɛ, ɛ 1,..., ɛ k, and ɛ, for 2 N so that γ M γ r ɛ, ξ M j ξ j r ɛ j, and a M a r ɛ, 47 for 1 j k and 2 N. In the present work we compute the approximate Fourier coefficient sequences {γ m } M m= M, {ξ j m } M m= M for the periodic orbit/parameterization of the normal bundles, as well as the constants r and ɛ 1, ɛ 1,..., ɛ k, using the computer aided methods of [14, 15] for the periodic orbit and the methods of [16, 17] for the basis functions of the normal bundles. The approximate Fourier coefficients {a,m } M m= M all in Cd for the solutions of the homological Equations are computed numerically using the methods of [1]. What remains is to compute validated constants ɛ for 2 N satisfying 47. This is topic of Section 3. We note that all of this data is assumed as input for the validation algorithm above. 3 Rigorous solution of the homological equations The goal of this section is to obtain computer assisted error bounds satisfying 47 on the solutions a : A r C d of the homological equations for 2 N. Or to put it another way, the goal is to obtain the data hypothesized in Assumption 3 of Section 1.1. More precisely, recall that for each N k we seek a so that da w dw where Λ def = 1 λ k λ k, and with fp w, z = For j = 1,..., d, denote with Denote a = a 1 Similarly, denote with F = F 1 We look for a solution of F j,m + Λa w = f P w, f P wz = = def = 2πim T,..., a d R d 2T = m Z f P,m e 2πi 2T mw z. + Λ a j,m f P j,m. 48 a = a N =2 = aj j=1,...,d, =2,...,N and F = F N =2 T,..., F d R d and a j = a j,m, for j = 1,..., d. m Z j = F j=1,...,d, =2,...,N F j = F,m j, for j = 1,..., d. m Z F a =

23 Given a bi-infinite sequence c = c m m Z of complex numbers and given ν 1, let c 1,ν def = c m ν m, m Z which is used to define the Banach space Denote Bl 1 ν, l 1 ν def = { l 1 ν A : l 1 ν l 1 ν def = {c = c m m Z : c 1,ν < }. } A is linear and A def = sup Av 1,ν < v 1,ν=1 the space of bounded linear operator from l 1 ν to l 1 ν. For each N k with {2,..., N} and j = 1,..., d, we consider a j n = nm, N be the cardinality l 1 ν. Let n = # { = 1,..., k N k = k {2,..., N} }. For each i = 1,..., n, denote by i N k the corresponding multi-index such that i {2,..., N}. In other words, we choose an ordering for the set of multi-indices satisfying {2,..., N}. Consider the Banach space X def = l 1 nd ν 5 endowed with the norm a X = { } max i=1,...,n j=1,...,d a j i 1,ν. 51 The goal is to compute solutions of 49 within the Banach space X, X using a Newton-Kantorovich type theorem the radii polynomial approach, e.g. see [65, 14]. This approach first requires computing an approximate solution using a finite dimensional projection. Given a Fourier truncation order M, consider a finite dimension projection Π M : X = l 1 ν nd R nd2m 1 defined by Π M : l 1 ν nd R nd2m 1 : a Π M a = a j i,m. m <M i=1,...,n j=1,...,d Given a nd, l 1 def ν denote af = Π M a. Consider the finite dimensional projection F m : R nd2m 1 R nd2m 1 of 49 given by F M a F = n F M i a F i=1 def = Π M F a F R nd2m 1. Assume that ā F is an approximate solution, that is that F M ā F. In practice we use a numerical Newton scheme. For the sake of simplicity, we introduce the notation ā lν 1 nd to denote the embedding of the numerical approximation āf R nd2m 1 by adding zeroes in the tail. 23

24 The idea now is to introduce a Newton-like operator based at ā whose fixed point correspond to a solution near ā of the infinite dimensional problem 49. A Newton-like operator is an operator of the form T a = a AF a, 52 with A a linear operator chosen carefully so that it is a good enough approximate inverse for the derivative operator DF ā. Since the Newton operator a a DF a 1 F a should be a contraction on a neighborhood of a fixed point, there is hope that for a carefully chosen approximation A of DF ā 1 the perturbation of the Newton operator 52 is itself a contraction near the fixed point. The choice of the operator A is problem dependent, and by using carefully the structure of the problem one can get better approximate inverses. For the specific problem 49 there is natural choice which we present next. 3.1 Construction of the approximate inverse A Consider the Jacobian matrix DF M ā M nd2m 1 C and let A M a numerically computed pseudo-inverse of DF M ā. Denote A M block-wise as A M 1,1 1,1 A M 1,1 1,d A M 1,n 1,1 A M 1,n 1,d A M 1,1 d,1 A M 1,1 d,d A M 1,n d,1 A M 1,n d,d A M = M nd2m 1 C A M n,1 1,1 A M n,1 1,d A M n,n 1,1 A M n,n 1,d A M n,1 d,1 A M n,1 d,d A M n,n d,1 A M n,n d,d 53 with A M n 1,n 2 d1,d 2 M 2M 1 C for 1 n 1, n 2 n and 1 d 1, d 2 d. We describe the operator A which serves as the approximate inverse for Df x. Denote by c F = c M+1,..., c M 1 C 2M 1. Given b = b i n i=1 = bj i i=1,...,n X, the action j=1,...,d of the operator A on b is given by Ab j1 i 1 = n i 2=1 Ai1,i 2 b i2 j1 = n i 2=1 j 2=1 d A i1,i 2 j1,j 2 b j2 i 2, where the action of A i1,i 2 j1,j 2 on c l 1 ν is A i1,i 2 j1,j 2 c m = δ i1,i 2 δ j1,j 2 A M i 1,i 2 j1,j 2 c F 1 2πim 2T + i 1 Λ m, m < M, c m, m M, 54 24

25 and where δ i,j equals 1 if i = j and otherwise. A has block-wise form given by A 1,1 1,1 A 1,1 1,d A 1,n 1,1 A 1,n 1,d A 1,1 d,1 A 1,1 d,d A 1,n d,1 A 1,n d,d A = A n,1 1,1 A n,1 1,d A n,n 1,1 A n,n 1,d A n,1 d,1 A n,1 d,d A n,n d,1 A n,n d,d 55 where A n1,n 2 d1,d 2 B l 1 ν, l 1 ν for 1 n1, n 2 n and 1 d 1, d 2 d. For c = c m m Z l 1 ν. Lemma 3.1. Define T as in 52 with A defined by 55. Then AF : X X. 56 Proof. Let h X, b def = F h and a def = Ab. Denote b = b i n i=1 = bj is to show that a X. We have i i=1,...,n j=1,...,d. The goal a = { } n a i i=1 { } = a j i i=1,...,n j=1,...,d { } = Ab j i i=1,...,n j=1,...,d { n } j = Ai,i2 b i2 i 2=1 n = i 2=1 j 2=1 Fix i {1,..., n} and j {1,..., d}, and let ci, j def = d A i,i2 j,j2 b j2 n i 2=1 j 2=1 i=1,...,n j=1,...,d i 2 i=1,...,n j=1,...,d d A i,i2 j,j2 b j2 i 2. Let us show that ci, j l 1 ν. Recalling 87, 48 and that b = F h, we have b j2 i 2 =. 25

26 F j2 i 2 h, and therefore ci, j 1,ν = = + + = + + m= m= m= M+1 ci, j m ν m M 1 2πim 2T + i Λ F j M 1 n d A M i 1,i 2 j1,j 2 i 2=1 j 2=1 1 2πim 2T + i Λ F j M [2πim 1 2πim 2T + i Λ 2T M 1 n d A M i 1,i 2 j1,j 2 i 2=1 j 2=1 [2πim 1 + i Λ 2T m=m m= m= M+1 m=m 2πim 2T i m h ν m F j2 i h 2 F i m h ν m ν m m ] + i Λ h j i,m f P j i,m h ν m F j2 i h 2 F ν m m ] + i Λ h j i,m f P j i,m h ν m, where the middle sum is finite, and the first and the third sums are bounded since h j i l1 ν, and since by the Banach algebra property of l 1 ν, we have that f P j i h l1 ν. Therefore, ci, j 1,ν < for all i {1,..., n} and j {1,..., d}. Hence, a = AF h X. We introduce the notation conjz to denote the complex conjugate of a complex number z C. Since we are interested in real solutions, we impose the following symmetries on the blocks of A. Assume that A M satisfies A M i 1,i 2 j1,j 2 k1, k 2 = conj A M i 1,i 2 j1,j 2 k1,k 2, k 1, k 2 = M + 1,..., M Define the space def X sym = l1 ν nd, 58 where l 1 ν def = { c l 1 ν c m = conjc m m Z }. 59 Denote by B r ā the closed ball of radius r > centered at ā in the Banach space X, that is B r ā = {a X : a ā X r}. Lemma 3.2. Assume that ā X sym and set r >. Define T as in 52 and let A defined by 55 such that the assumption 57 holds. Then AF : X sym X sym. 6 Assume moreover that T : B r ā B r ā is a contraction, and let ã X denote the unique fixed point of T in B r ā which exists by the contraction mapping theorem. Then, ã X sym. 26

27 Proof. We begin by showing that 56 holds. Let h X sym, b def = F h and a def = Ab. As in the proof of Lemma 3.1, we have that n d a = A i,i2 j,j2 b j2 i 2. i 2=1 j 2=1 i=1,...,n j=1,...,d By Lemma 3.1, we know that a X <. It remains to show that each component a j i is in l 1 ν. Now, for each i, i 2 = 1,..., n and j, j 2 = 1,..., d, let c def = A i,i2 j,j2 b j2 i 2. Then, for each m Z, and using the symmetry assumption 57, c m = = m 1= k 1= A i,i2 j,j2 m,k1 b j2 i 2 k 1 A i,i2 j,j2 m, k1 b j2 i 2 k 1 = conj A i,i2 j,j2 m,k1 conj b j2 i 2 k 1 k 1= = conj A i,i2 j,j2 m,k1 b j2 i 2 k 1 k 1= = conj c m. Finally, by 56, T : X sym X sym. Using that ā X sym B r ā, and that X sym is a closed subset of X, we obtain that ã = lim n T n ā X sym. 3.2 The radii polynomial approach The goal of the present section is to describe an efficient way of determining a ball of the form B r ā on which the operator T : X X as defined in 52 is a contraction. For each i = 1,..., n and j = 1,..., d, consider bounds Y j i and Zj i satisfying j T ā ā Y j j i and sup DT ā + bc Z j i. 61 1,ν 1,ν i b,c B r Remark 3.3 The bound Z j i as a polynomial in r. To compute Zj i for i = 1,..., n and j = 1,..., d, one estimates DT ā + bc j i for all b, c B r. This is equivalent to estimating DT ā + urvr j i for all u, v B 1. If the nonlinearities of the vector field f are polynomials, then F will consists of Cauchy products of discrete convolutions. Since T x = x AF x and DT ā + urvr X, then each component of DT ā + urvr can be expanded as a polynomial in r with the coefficients being in l 1 ν. Hence, from now on we assume that the bound Z j i = Zj i r is polynomial in r. Definition 3.4. For each i = 1,..., n and j = 1,..., d, the radii polynomial p j i r is given by p j def i r = Y j i + Zj i r r. 62 i 27

28 Lemma 3.5. Define I = n d i=1 j=1 { r > : p j i r < }. 63 If I, then I = I, I + is an open interval, and for any fixed r I, the ball B r ā contains a unique solution of 49. The proof of this result is standard e.g. see [14, 65, 66] and will be omitted. Before proceeding with the explicit computation of the bounds satisfying 61, we present a remark describing how a successful application of Lemma 3.5 can be used to obtain the rigorous error bounds required in 47. Remark 3.6. Consider the radii polynomials 62 constructed with a fixed decay rate ν > 1. Let I the set defined in 63 and assume it is non empty. Let r I and let a B r ā such that F a =. Then { } a ā X = max i=1,...,n j=1,...,d a j i āj i 1,ν r. 64 Let be a multi-index such that 2 N. In the context of the error bounds 47, set a M = ā. Also, set r = T logν. 65 π the width of the strip A r in the complex plane.then, 2 N, using 64, a M a r = ā a r = max 1 j d āj = max 1 j d sup = max 1 j d sup ā j w A r w A r max 1 j d sup max a j A r w a j w ā j,me 2πim 2T m= w A r m= 1 j d m= max 1 j d m= = max ā j 1 j d ɛ. ā j ā j,m a j ā j a j,m a j,m,m a j,m w a j,m e π m T ν m,me 2πim 2T e πim T def Hence, the rigorous error bound required in 47 can be set to be ɛ = r for all 2 N. Here is our general strategy: we use Lemma 3.5 to solve 49 and Remark 3.6 shows how the rigorous error bounds required in 47 can be obtained. Next, we attempt to provide general formulas for the construction of the radii polynomials valid for any vector field, any periodic orbit, no matter what the dimension of the manifold and the dimension of the ν r w w 28

29 phase space are. However, some of the formula would be significantly hard to write in the full generality and almost impossible to read, hence we prefer to combine the exposition of the general case with a concrete example. To begin with, recall that the unknowns of the problem are the sequences a j l 1 ν of Fourier coefficients of the functions a j w where {2,..., N} and j = 1,..., d. Denote by ã j w with {, 1} the data for the problem: ã j is the sequence of Fourier coefficients on the basis of 2T -periodic functions of the periodic orbit γt while ã j with = 1 are sequences of the Fourier coefficients forming the normal bundle. We assume that both the periodic orbit and the linear bundle have been previously computed and that the data are given in the form ã j = ā j + Ea j, {, 1}, j = 1,..., d, 66 where ā j m for any m > M and the remainder Ea j is known only in norm, that is Ea j 1,ν ɛ j. For any a X, the function F a depends on the data ã with {, 1}. Let us write explicitly such a dependence as F a = F {ã } 1 =, a = F {ā + Ea } 1 =, a. Denote F a = F {ā } 1 =, a and introduce EF a so that the equality holds F a = F a + EF a. 67 In practice, EF a is a correction that can only be estimated in norm, since the contributions Ea j, {, 1}, j = 1,..., d are only known in norm. From now on, the notation E always stands for a quantity for which all we know is an upper bound for its norm. Similarly, the derivative DF a splits into the sum where DF a = DF {ā } 1 =, a Bound Y DF a = DF a + EDF a 68 Formally, for each i = 1,..., n and j = 1,..., d, j n d [ ] AF ā = j A i,i2 j,j2 F ā 2 j2 i i 2 + EF i 1,ν ā 2 i 2=1 j 2=1 1,ν n d j A i,i2 j,j2 F ā 2 n d i 2 + A i,i2 j,j2 EF j2 1,ν i 2=1 j 2=1 i 2=1 j 2=1 i 2 ā. 1,ν Any F ā j i is a finite dimensional sequence, the length of which depends on the degree of the nonlinearity of the vector field. Therefore the first sum is a finite computations which can be rigorously bounded using interval arithmetic. For the second sum we estimate n d A i,i2 j,j2 EF j2 i 2 ā n d A i,i2 j,j2 EF j2 i ā 2 1,ν 1,ν i 2=1 j 2=1 i 2=1 j 2=1 and we take advantage from the Banach algebra property a b 1,ν a 1,ν b 1,ν in order to bound EF j2 i 2 ā 1,ν. 29

30 3.2.2 The bounds Z j i, i = 1,..., n, j = 1,..., d For each i = 1,..., n and j = 1,..., d we seek a bound Z j i, a polynomial in the variable r satisfying j sup DT ā + bc Z j i r. 1,ν b,c B r Recalling that DF M ā M nd2m 1 Cā is the Jacobian of F M at ā, denote as DF M = {DF n M 1,n 2 d1,d 2 } the component-wise representation of DF M ā similar to 53. Define the linear operator A so that, for any b l 1 ν nd A b j1 i 1 = n i 2=1 i j1 n A i 1,i 2 b i2 = i 2=1 j 2=1 d A i 1,i 2 j1,j 2 b j2 i 2, where the action of A i 1,i 2 j1,j 2 on h l 1 ν is given by DF M A i 1,i 2 j1,j 2 h F, m < M, m i 1,i 2 j1,j 2 h = m 2πim δ i1,i 2 δ j1,j 2 2T + i 1 Λ h m, m M, 69 where δ i,j equals 1 if i = j and otherwise. Consider now the splitting DT ā + bc = I ADF ā + bc = I AA c ADF ā A c ADF ā + b DF āc and DT ā + bc j i j I AA c + ADF ā A c 1,ν i 1,ν j + ADF ā + b DF āc. 1,ν i j i 1,ν Since b, c B r, we can factor out r and write b = ru and c = rv where u, v B 1. This means that u = {u j i } i=1,...,n satisfies u j j=1,...,d i 1,ν 1 for any i = 1,..., n and j = 1,..., d. The same holds for v. Then Z j i r = Z j i r + Z1 j i r + Z2 j i r2 + + Z p j i rp where I AA v ADF ā A v ADF ā + ru DF ārv j i j i j i Z j i, v X 1 1,ν Z 1 j i, v X 1 1,ν Z 2 j i r2 + + Z p j i rp, u X 1, v X 1. 1,ν The exponent p refers to the maximal power of r that appears on the last relation and equals the degree of the polynomial nonlinearity of the vector field fx. 3

31 3.2.3 The bound Z Denote by B def = I AA. Since the tail diagonal components of A i,i j,j are defined as the inverse of the tail diagonal components of A i,i j,j, the operator B results in a finite dimensional operator acting on v F. Hence j I AA v = Bv j i = i n d i 2=1 j 2=1 B i,i2 j,j2 v F j2 i 2. The operator DF M used in the definition of A represents the genuine finite-dimensional Jacobian of F. Therefore DF M and A inherit the same splitting already discussed for DF, that is A = Ā + EA. Thus, we set B = B + EB where B = I AĀ and EB = AEA. It follows j I AA v j Bv i j 1,ν + AEA v 1,ν i n i 2=1 j 2= The bound Z 1 ADF ā A v d B i,i2 j,j2 + j i n i 1,ν i 2=1 j 2=1 d A i,i2 j,j2 Z 1 j i, v X 1. 1,ν EA v j2 In order to properly compute a bound of the above norm, we need to recast the quantity ADF ā A v as a linear operator acting on the components of v. It is convenient to treat the linear operator DF ā A as a matrix of operators. Recall that i {1,..., n} and j {1,..., d}. Without separating further the different role of the two indexes, we can consider a unique index ranging from 1 to nd labelling the elements of v. We represent the operator DF ā A by the block operators of the form Γ = Γm, m Bl 1 ν, l 1 ν with m, m {1,..., nd}. Moreover, for each m {1,..., nd} [ DF ā A v] m = nd m=1 Γm, mv m. We also represent A by the operator blocks Am, m with m, m {1,... nd}. Then the operator ADF ā A is represented by AΓ with and for each q {1,..., nd} Therefore nd AΓm, m = Am, qγq, m Bl 1 ν, l 1 ν q=1 ADF ā A v = q nd m=1 AΓq, mv m. i 2. 1,ν 31

32 ADF ā A v 1,ν q nd m=1 nd m=1 m=1 AΓq, mv m 1,ν nd Aq, mγm, mv m 1,ν. Also, we separate the E-contribution and write Γm, m = Γm, m + EΓm, m. For each q {1,..., nd}, we can finally define Z 1 q as ADF ā A v 1,ν q nd nd m=1 m=1 nd nd m=1 m=1 + =: Z 1 q. nd m=1 m=1 Aq, mγm, m + EΓm, mv m 1,ν Aq, mγm, m nd Aq, m EΓm, mv m 1,ν The previous formula provides the scheme for the computation of the bound Z 1 for any given vector field. However, the definition of Γ is problem dependent, hence we prefer to present more details about the construction of the bound Z 1 having a concrete case in mind. That will be done in the subsequent section, where the Lorenz system is considered Bound Z i, 2 i p For the construction of the bound Z i, 2 i p we proceed as follows: define a vector of sequences {V q } nd q=1, V q = {V q m } m Z so that V q contains the r i contributions of [ DF ā + ru A rv ]. Then introduce V = {V q q} nd q=1 so that V q sup u X 1, v X 1 V q 1,ν. Finally, define Z i q = nq m=1 Aq, m V m. 3.3 Enclosure of a in case of large N The first condition 28 required by Theorem 2.4 is that N +1 > κ/µ, where κ only depends on the Jacobian of the vector field along γ and µ on the Floquet exponents. If the degree of the polynomial nonlinearity of the vector field is large, or the magnitude of the orbit γ is large in phase space, the bound κ can be considerably large. Also, if the real part of the Floquet exponents are small, µ is small. The combination of these factors may lead to a choice of large N. From the theoretical point of view, the procedure works for any N. On the other hand, the number of equation in the nonlinear system F,m j = to be rigorously solved increases with N and the effective computation is sometimes problematic. In this section we present a modification of the technique which facilitates computation of the coefficients a when is large. The key observation is that a r for large enough, thus it is reasonable to compute the enclosure of a around ā =. Suppose that for a choice of Ñ < N the method of the previous sections returns the enclosure of the functions a w for Ñ. We are now concerned with the computation

33 j of a w for Ñ < N, that is, we need to solve F,m =, in the unknowns a j w, Ñ < N. Instead, a with Ñ are data of the problem and given within rigorous bounds computed before. Since F depends on a and on a β with β we can solve layer-by-layer. That means, we solve {F = } = Ñ+1 in the unknown {a } = Ñ+1. Once the Ñ + 1 layer is rigorously enclosed, we move to the next Ñ + 2 layer, treating the previous computed coefficients as data for the new problem. We apply the same technique as before, with the following modifications. The numerical approximate solution ā is set to zero. For a choice of M, that might be different than the previous, define ā j,m = for any m < M, j = 1,..., d, Ñ < N. The operators A and A, see 87, 69 are now defined as diagonal operators in, that is A i1,i 2 if i 1 i 2. The same for A. In practice only the Jacobian of F with respect to a is considered in constructing A and A, even in the finite dimensional subspace. Use the notation A, in place of A i,i when = i. The definition of the Z bound is the same as before. However, since A and A are diagonal in, the operators Γm, m are slightly different and the sum 7 is taken over those m s that refer to the same as q. The meaning of these statements will be clearer in the next section, where the computational technique is applied to the Lorenz system. 4 Unstable and unstable manifolds of periodic orbits in the Lorenz equations As a first application of our proposed method, we rigorously compute some invariant manifolds attached to periodic orbits in the Lorenz equations u = fu, u = x, y, z where f is the vector field given by fx, y, z = σx + σy ρx y xz βz + xy βa 3, σ, β, ρ R. 72 The composition of f and the parameterization P reads as σa 1 w + σa 2 w f P w = ρa 1 w a 2 w a 1 a 3 w w + a 1 a 2 w, where a 1 a 3 w def = a 1 1 wa 3 2 w = i and a 1 a 2 w def = a 1 1 wa 2 2 w = i Moreover, f P,m = σa 1,m + σa 2,m ρa 1,m a 2,m a 1 a 3,m βa 3,m + a 1 a 2,m, 33

34 where a 1 a 3 = a 1 1 a 3 2,m m = i and a 1 a 2 = a 1 1 a 2 2,,m m = i with a 1 1 a 3 2 m = k 1 +k 2 =m k i Z a 1 1,k 1 a 3 2,k 2 and a 1 1 a 2 2 m = k 1 +k 2 =m k i Z a 1 1,k 1 a 2 2,k 2. Therefore, in the context of the Lorenz system, 48 is given by F 1 2πim,m 2T F,m 2 = + λ a 1,m + σa 1,m σa 2,m 2πim 2T + λ a 2,m ρa 1,m + a 2,m + a 1 a 3,m 2πim 2T + λ a 3,m + βa 3,m a 1 a 2,m F 3,m with 2 N and m Z. Concerning the splitting 67, it follows EF a 2 = EF a =, 73 Ea 1 a Ea 3 a Ea 1 1 a Ea 3 1 a Ea 1 1 Ea 3 1 Ea 1 a Ea 2 a Ea 1 1 a Ea 2 1 a Ea 1 1 Ea 2 1 Ea 1 a 3 + Ea 3 a 1 + Ea 1 1 a Ea3 1 a 1 1 Ea 1 a 2 + Ea 2 a 1 + Ea 1 1 a Ea2 1 a 1 1 Since l 1 ν is a Banach algebra with the convolution product, it follows that EF a 1,ν, > 2. ɛ 1 a3 1,ν + ɛ 3 a1 1,ν + ɛ 1 1 a3 1 1,ν + ɛ 3 1 a1 1 1,ν + ɛ 1 1 ɛ3 1 δ,2 ɛ 1 a2 1,ν + ɛ 2 a1 1,ν + ɛ 1 1 a2 1 1,ν + ɛ 2 1 a1 1 1,ν + ɛ 1 1 ɛ2 1 δ,2 The derivative of F at any point a is a linear operator DF a whose action on v l 1 ν 3n is explicitly given by σv,m 1 σv,m 2 2πim DF av,m = 2T + λ v,m + ρv,m 1 + v,m 2 + v 1 a 3 + a 1 v 3,m,m where v i a j,m = = i, 1 2 v i 1 a j = 2 m βv 3,m = i, 1 2 v 1 a 2,m a 1 v 2,m 74 a j 2 v i 1 = a j v i. m,m For any a, the derivative DF a splits into two parts 68. The data a and a 1 contribute only to the nonlinearity, therefore we have EDF av = Ea 1 v 3 + Ea 3 v 1 + Ea 1 1 v Ea3 1 v 1 1. Ea 1 v 2 + Ea 2 v 1 + Ea 1 1 v Ea2 1 v ,

35 Hence, for any v B 1 EA v 1,ν 2 ɛ 1 + ɛ 3 ɛ 1 + ɛ 2, EA v 1,ν ɛ 1 + ɛ 3 + ɛ ɛ 3 1 ɛ 1 + ɛ 2 + ɛ ɛ 2 1, for all > 2. Now we present in more details the definition of Z 1 satisfying 71. For x = {x m } m Z we define x I = I Π M x, that is the sequence so that x I m = for m < M, x I m = x m for m M. Our goal is to write the action of DF ā A. Recalling formula 74, the definition of A and the assumption 66, it turns out that [ DF ā A v ],m = ā3 v 1I,m + ā 1 v 3I,m ā 2 v 1I,m ā 1 v 2I,m σv,m 1 v,m 2 ρv,m 1 + v,m 2 + ā 3 v 1 +,m βv,m 3 ā 2 v 1,m + Ev,m, ā 1 v 3,m ā 1 v 2,m m < M + Ev,m, m M. In the term Ev,m all the contributions due to Ea and Ea 1 are collected. In order to define the matrix of operators Γ, it is convenient to introduce further notation. Let Ĩ the infinite dimensional matrix given component wise by { δj,m, j, m M Ĩj, m =, otherwise. To any ā j we associate the matrices A j and Ãj A j m, m = ā j m m, Ã j m, m = as {, m, m < M ā j m m, otherwise. 75 The action of Ĩ on a sequence w = {w m} m Z is Ĩw m = for m < M and Ĩw m = w m for m M. The action of A j on w is such that A j w m = ā j w m while Ãj w m = ā j w I m for m < M and Ãj w m = ā j w m for m M. We are now in the position of writing σĩv1 v 2 DF ā A v = ρĩv1 + Ĩv2 + βĩv = i, = i, 1 2 Ã 3 2 v Ã1 2 v 3 1 Ã 2 2 v Ã1 2 v Ev. Referring to the label set {1,..., nd} previously introduced, we can now define the operators Γs, t associated to DF ā A, for any s, t {1,..., nd}. Suppose the couples of labels i, j, i {1,..., n}, j {1,..., d} are one-to-one related to the set {1,..., nd} through the bijection φ. Given q = φi, j {1,..., nd} denote by q = i and q j = j. For instance [ DF ā A v ] s = [ DF ā A v ] s j s. Also, given q = φi 1, j 1 and s = φi 2, j 2, according to the notation in 55, Aq, s = A i1,i 2 j1,j 2. 35

36 The non zero Γs, t are the following: t s j = 1 s = t j = 1 Γs, t = σĩ t j = 2 Γs, t = σĩ t j = 1 Γs, t = ρĩ + Ã3 s j = 2 s = t t j = 2 Γs, t = Ĩ t j = 3 Γs, t = Ã1 t j = 1 Γs, t = Ã2 s j = 3 s = t t j = 2 Γs, t = Ã1 t j = 3 Γs, t = βĩ t, s > t j = 1 Γs, t = Ã3 s t t j = 3 Γs, t = Ã1 s t t, s > t j = 1 Γs, t = Ã2 s t t j = 2 Γs, t = Ã1 s t Concerning the contribution of EΓ, as already mentioned before, we are interested in the the image of EΓs, t applied to a sequence w l 1 ν: s j = 2 s = t t j = 1 EΓs, tw = Ea 3 w t j = 3 EΓs, tw = Ea 1 w, s = t + 1 s j = 3 s = t t j = 1 EΓs, tw = Ea 2 w t j = 2 EΓs, tw = Ea 1 w, s = t + 1 t j = 1 EΓs, tw = Ea 3 1 w t j = 3 EΓs, tw = Ea 1 1 w t j = 1 EΓs, tw = Ea 2 1 w t j = 2 EΓs, tw = Ea 1 1 w Therefore, the various terms EΓs, tv t 1,ν appearing in 71 are easily bounded by either Ea i 1,ν = ɛ i or Ea i 1 1,ν = ɛ i 1. What remains to do it to provide a way to compute Aq, sγs, t. Looking at the definition of A, we realize that Aq, s is either a square finite dimensional matrix of dimension 2M 1 or an infinite dimensional operator with a diagonal action out of the central block. The latter case holds when q = s. We depict the two cases... Aq, s = A, for q s and Aq, q = A Suppose Γs, t = Ãj the case Ĩ is immediate. Since āj m = for m M, the infinite dimensional matrix Γs, t is zero out of a strip 2M 1 thick around the main diagonal. For what follows, it is useful to introduce the matrix ā ā ā M ā 1 ā ā 1... ā M+2 ā M E = ā M ā ā M+1 ā M ā ā M ā M ā and the M 1 M 1 sub matrices ˆB, Ĉ of E as Ĉ E =. ˆB 36

37 Finally, define B and C the 2M 1 2M 1 matrices given by B = ˆB and C = Ĉ Using these matrices, we can see the infinity dimensional operator Γs, t as tridiagonal concatenation of B, E, C with a empty block in the centre, that is E C B E C Γs, t = B C. 77 B E C B E C In case Γs, t is Ĩ, then E = I and B = C =. Let us now compute the multiplication Aq, sγs, t. If p q the operator Aq, s is of the first of the two forms depicted in 76. It follows that Aq, sγs, t = AB AC, which is a finite dimensional operator and the operator norm can be easily computed. Note that if Γs, t = cĩ then Aq, sγs, t =. Consider now the case s = q. The operator A is of the form... D3 D 2 D 1 Aq, q = A, H 1 H 2 H 3... where dimd i = dima = dimh i = 2M 1 2M 1. Remember that the elements on the diagonal are strictly decreasing, hence max m D i m, m < inf m D i 1 m, m and the same for H i. The multiplication with Γs, t produces 37

38 As, sγs, t = D 3 B D 3 E D 3 C D 2 B D 2 E D 1 B }{{} V 3 }{{} V 2 }{{} V 1 D 2 C D 1 E D 1 C AB AC H 1 B H 1 E H 2 B } {{ } M H 1 C H 2 E H 2 C H 3 B H 3 E H 3 C }{{} W 1 }{{} W 2 }{{} W 3. The operator norm of the above infinite dimensional operator is given by As, sγs, t = max{ M, max{ V i }, max{ W i }}. Since D i < D i 1 and H i < H i 1, it follows that V i V i 1 and W i W i 1. Hence we conclude that As, sγs, t max{ M, V 1, W 1 }. Such a quantity can be rigorously computed because it involves only a finite number of operations. Concerning the Z 2 bound, in the case of the Lorenz system, the r 2 contribution is DF ā + ru A rv u 1 = 1 v u 3 1 v 1 2 r = i u 1 1 v u 2 1 v = i 2 Since u j 1 v l 2 1,ν u j 1 1,ν v l 2 1,ν 1, we define { if q V q = j = 1 2q 3 if q j = 2 or q j = 3, and q Computation of the higher order terms a j, for Ñ < N Now we apply the variant described in Section 3.3 to the case of the Lorenz system. Suppose the enclosure of the functions a j has been computed for all Ñ. It follows that a j = ā j + Ea j with Ea j ɛ j, for any Ñ and j = 1, 2, 3. We are concerned with the computation of añ+1, the solution of FÑ+1 =. Let = Ñ + 1. The system F is formally the same as 73. It is however convenient to rewrite the nonlinearity in the form a 1 1 a 3 2 = a 1 1 a a 1 a 3 + a 1 a 3, = i = i Ñ 38

39 where the dependence on the unknown a is highlighted. Setting ā =, it follows that F ā = = i Ñ = i Ñ a 1 1 a 3 2 a 1 1 a 2 2. A bound for F ā 1,ν is provided by F ā 1,ν = i Ñ = i Ñ 1 ā ā 2 2 ā 1 ā 1 1,ν 1,ν = i Ñ = i Ñ ā 1 1 1,ν ɛ ā 3 1 1,ν ɛ ɛ 1 1 ɛ 3 2. ā 1 1 1,ν ɛ ā 2 1 1,ν ɛ ɛ 1 1 ɛ 2 2 The definition of A, and A, are based on the Jacobian of F with respect to a, say DF,. Recalling the definition 75 and denoting by µ the diagonal matrix with µ m, m = 2πim 2T + λ m, the operator DF, = DF, + EDF,, where µ + σi σi DF, = ρi + A 3 µ + I A 1 A 2 A 1 µ + βi EDF, v 1 v 2 v 3 = Ea 3 v 1 + Ea 1 v3 Ea 2 v 1 Ea 1 v2 The finite dimensional operator A, M is defined as approximate inverse of DF, M, while A, M is given by DF, M. The operators Γs, t are similar to those reported in the previous section. The difference is that any Ãj with > is replaced by zero. The construction of Zq 1 follows form 7. However, since the only unknown is a, the sum in 7 restricts to those s such that s =. Explicitly, for j = 1, 2, 3,. Z 1,j 2 A, j,1 σĩ + A, j,2 ρĩ + Ã3 + A, j,2 Ĩ + A, j,2 Ã 1 + A, j,2 ɛ 1 + ɛ 3 + A, j,3 Ã2 + A, j,3 Ã1 + A, j,3 βĩ + A, j,3 ɛ 1 + ɛ 2. We remark that some of the above terms vanish, for instance A, j,i Ĩ = whenever j i. 39

40 4.2 Some remarks 1. According to 47, the computation of the parameterization begins with the rigorous enclosure of the periodic orbit γt. We use the method described in [14] in order to rigorously enclose the Fourier expansion of γ. Denoted by T the period of the orbit, the function γt is expanded on the T -periodic exponential Fourier basis. For a choice of ν γ, the sequence of Fourier coefficients γ = {γ m } m Z is proved to be in a ball of radius r γ, with respect to the ν γ -norm, around the numerical approximation γ M. Define ν < ν γ and r as in 65. Arguing as in Remark 3.6, it follows that γ M γ def r ɛ, where ɛ = r γ. 2. The computation of the normal bundle is performed following the method described in [16]. The functions ξ j w are defined as ξ j w = Qwv j, where v j is an eigenvector of R and Qw, R is the Floquet normal form decomposition of the fundamental matrix solution of the linearized system around γt. The matrix function Qt is expanded in Fourier series on the 2T -periodic exponential basis. Denoted by Q the sequence of Fourier coefficients of Qw, the computation returns a radius r F l so that Qi, j Qi, j 1,ν r F l, Ri, j Ri, j r F l, 1 i, j d where Q, R is a finite dimensional approximate solution. Without loss of generality, let us choose j = 1 and denote v = v 1, so that ξ 1 = Qwv. The eigenvector v of R is computed rigorously, for instance following [67], so that vj vj r v, for any 1 j d. For convenience, let us write Q = Q + ɛ Q and v = v + ɛ v, with ɛ Q 1,ν r F l, ɛ v r v both component wise. Set ξ 1 = Q v. Thus and ξ 1 ξ 1 = Q + ɛ Q v + ɛ v Q v = ɛ Q v + Qɛ v + ɛ Q ɛ v ξ 1 i ξ 1 i 1,ν ɛ Q i, j vj 1,ν + Qi, jɛ v j 1,ν + j j j r F l vj + r v Qi, j 1,ν + dr F l r v. j j j ɛ Q i, jɛ v j 1,ν Hence max ξ 1i ξ 1 i 1,ν r F l vj + dr F l r v + r v max Qi, j 1,ν. i=1,...,d i Again, according to Remark 3.6, the bound ɛ 1 appearing in 47 is provided by the right hand side of the above relation. The same applies for all the normal bundles ξ i, 1 i k. In the case of the Lorenz system d = The eigenvalues λ of R are also given within some bounds, precisely λ λ r v. 4. The parameterization of the normal bundle, that is a w with = 1 is defined in terms of the eigenvectors v j of the matrix R. Therefore the scaling of the eigenvectors is a further free parameter. From one side a larger eigenvectors will result in a larger image of the parameterization. On the other, the rigorous enclosure of the coefficients might be problematic. Besides the scaling of the eigenvectors, the size of the image of the validated parameterization is affected by the choice of the Taylor norm decay parameter ν. j 4

41 4.3 Validation values for the Lorenz system We now report the validation values for the stable manifold of a periodic orbit of the Lorenz system, with σ = 1, β = 8/3. Fix the parameter ρ = 22 and choose the decay rate ν γ = 1.3 and ν = The periodic orbit is computed with period T and enclosure radius r γ = around a numerical approximate solution. Hence define ɛ = The computation for the Floquet normal form decomposition Q, R returns the enclosure radius r F l = The periodic orbit is hyperbolic has one positive and one negative Floquet exponent. Here we are concerned with the stable manifold. The stable eigenvalue λ and associated eigenvector v of R are proved to be in a ball of radius r v = around the numerical approximation λ = and v = [ , , ] T. The procedure explained in the second remark in Section 4.2 yields ɛ 1 = Now choose N = 1 and M = 4, respectively the order of the parameterization P N and the finite dimensional projection coefficient. The rigorous enclosure of the functions a j returns the bound ɛ = Following the validation algorithm proposed in Section 2.5, the validation values are defined as follows. First define r = and ν = 1. Note that the latter ν refers to the norm in the Taylor space. We need to find κ satisfying 21, that is Dfγ r κ. For the Lorenz equations, σ σ Dfγ = ρ γ 3 1 γ 1 γ 2 γ 1 β and γ = Γ + E γ where Γw = M m= M Γ me i 2π T mw and E γ r ɛ. Explicitly, from 17, Dfγ 3 r = max 1 i 3 j=1 Dfγi, j A r. In practice, one computes Dfγ A r = Df m Γ e 2πr T m + ɛ ɛ m <M ɛ ɛ and then takes the maximum over sum along the rows. In our case, we get κ = For the computation of C satisfying 22, let us first compute the norm a A r. For = 1,..., N, a w = ā + Ea with Ea r ɛ. Hence a A r m ā m e 2πr ˆT m + ɛ. The matrices A have the form A = a 3 a 1 a 2 a 1 Plugging in the r-norms above computed, and afterwards taking the maximum of the sums along the rows, we obtain A r A r

42 Inserting into formula 31, having λ = λ 1 [ r v, r v ], Ñ = N because the nonlinearity of the vector field is quadratic, it follows that C = The constants M 1, M 2 and µ satisfying 24, 25 and 23, respectively, are given by M 1 = 2, M 2 = 1 and µ = Introducing ρ satisfying 26 and ρ = 1.1ρ is not necessary in this example, because the second derivatives of the vector field are constants and do not depend on ρ. What remains to be computed is the constant ɛ satisfying 27, that is E N r,ν ɛ. By definition, E N is given by E N w, z = N+1 R wz, where for the Lorenz equations, a 1 1 wa 3 2 w R w = = i >. 79 a 1 1 wa 2 2 w = i > Since a w = for any > N, we get that R w = for any > 2N. Thus E N w, z = 2N =N+1 R wz and E N w, z r,ν max 2N 1 i 3 =N+1 The functions a i are known with explicit error bounds, that is a i r The norm of the components of 79 satisfy R r = N 1= N N 1= N R i r ν. ā i r + ɛ. ā 1 1 r + ɛ 1 ā 3 1 r + ɛ 1. 8 ā 1 1 r + ɛ 1 ā 2 1 r + ɛ 1 For the example under investigation, we obtain ɛ = Then we check that 28 holds, that is we check that N + 1 > κ/µ and define δ = so that the relation 3 is satisfied. Since also 29 is satisfied, we conclude that the finite order parameterization P N w, z is rigorously validated and the N-tail is such that H r,ν δ. Figure 2 shows the image of the parameterization P N w, z, z < ν above discussed. In Table 1 we summarize the data and results for other examples. Each line reports the value of the parameter ρ of the Lorenz system, the period T of the orbit, the finite order/dimension parameters N and M, the Fourier/Taylor norm parameters r, ν and the resulting δ so that H r,ν δ. All the computations concern the stable manifold. Figure 2, on the right, shows the image of the parameterization computed in the third example. We also compute the parameterization for the local unstable manifold for the periodic orbit with ρ = 28. The unstable Floquet exponents is λ = and a preliminary analysis shows that a value for N larger than 5 is required. We choose N = 7. Thus, setting Ñ = 1, M = 6, the computation of a, Ñ is performed according to the general technique. Rather, the remaining coefficients a, Ñ < N are enclosed layer-bylayer as discussed in section 4.1. In this case, we scale the eigenvector to be of length 2 and we set the Taylor decay rate ν = 1. The N-order parameterization is validated with N-tail norm H r,ν , being r = In Figure 3 on the right, the image of the unstable local manifold is plot. 42

43 Figure 2: Two images of the parameterization PN w, z, z ν, N = 1, of the local stable manifold associated to the periodic orbit of the Lorenz system at parameter ρ = 22. In the 11 left case ν = 1 and the N -tail norm is khk. On the right ν = 4 and r,ν khkr,ν ρ T N M r ν δ Table 1: Data for invariant manifolds of some periodic orbits in the Lorenz equations. Figure 3: Image of the local stable left and unstable right manifold associated to a periodic orbit of the Lorenz system at parameter ρ = A three-dimensional stable manifold of a periodic orbit in a suspension bridge equation In this section, we apply our proposed approach to rigorously compute a three-dimensional local stable manifold of a periodic orbit in a fourth order ODE see 82 below. The solu43